Now all we have to do, for a given prime pair (p,q), is to
find the roots of these bizarre polys, mod n, using
polrootsmod and the CRT.

Up to effete signs, we have solved the Lucas problem.
Up to effete signs, we have solved all the additional
Fermat/Euler bolt-ons, mod p. All that is left are the
Fermats mod q. As is clear from Arnault, you will
not need to spend long looping on p, k, m, to solve
Fermats mod q = 1 + 2*k*m*(p-1), for at least one
of the many chinese roots.

That's it. All that can happen is that the deviser of the
hopeless test may wriggle with gcds, and arbitrarily bolt on
additional Fermats/Euler/M-R's, thereby filling PrimeNumbers
with lots of noise that cannot change the fact that she/he is
bound to lose, because, in your own words:

> > fooled 10 times
> >
> > NB: Please, Paul, no more wriggles, sign tests, gcds, extra Fermats,
> > new choices of [P,Q], this August. The Gremlins are sunning
> > themselves and find it irkesome to tool up for such vain tests.
>
> Those are good counterexamples

Of course were you to add gcd(x^16+x^8-1,n)==1, in September,
the Gremlins would work with different cosines.

David

WarrenS

Tao, Harcos, Englesma, et al seem to have stalled trying to improve Zhang s upper bound of 70,000,000. They claim to have confirmed they got it down to 5414

Message 7 of 11
, Aug 4, 2013

0 Attachment

Tao, Harcos, Englesma, et al seem to have stalled trying to improve Zhang's upper
bound of 70,000,000. They claim to have confirmed they got it down to 5414 but look
like they aren't going to be able to go much further (perhaps can push it a bit below 5000
if combine all their juice?).